Solidify issue at sharp bifurcation location
You have two main issues.  The first is that you have a transition point which is joined by a single vertex.



That one vertex will cause you problems with several areas of Blender. Sculpting, Rendering and generative Modifiers to name a few.

There is no face at that center point and I imagine this is what's causing the Solidify Modifier to fail.

The other problem with this model is that the quads have all been converted to triangles and the loops are sort of spiraling around so un-triangulation is certain to fail because there are no loops with clearly defined axis.

Without quads that have well-defined loops, some important modeling tools will not work. So to start with we should try and avoid this as best as we can with the repairs to follow.  

You can start by selecting all the vertices around the edges and use 'Make Face'f so there is a well defined face to start working with.  You can do this by using Alt+'Mouse-Select-Click' on one of the boundary edges.  In this case it will select both perimeter loops since they are attached but you can simply unselect the second loop to get started.  This will be easier to do in wireframe mode using the Circle-Select toolc and 'Menu-Select-Click'(this will un-select).

Now with the new face selected use 'Extrude'e to extend the edges out into some extra faces.  Repeat this for the other side, select both faces and use xthenf for 'Delete->Faces'.



Now you can also run 'Remove Doubles' from the 'Specials Menu'w to clear up some of the overlapping Vertices.

Once you've done that you can select some of the edges which are close together and double tap g to slide them over to nearby edges.  If you have 'AutoMerge Editing' activated, the overlapping vertices that result from this will be removed automatically without you having to repeatedly use 'Remove Doubles'.  Using these tools you should be able to clean up many of the rendering errors that are visible along the boundaries of the model.  



That one vertex is still a problem but now it's only visible from the inside.  

I was not able to repair this without Applying the Solidify Modifier.
The modifier is causing the vertices at the junction to be difficult(each of them has a 'virtual position' as well and this is causing some logical conflicts some of the mesh editing tools).  If you want to clean that area a little more then you should Apply the Modifier.

You should consider looking for an import option that can maintain quads, it will allow you to use a lot more of Blender mesh editing tools.

UPDATE:  After trying again, I was able to repair the inside areas without applying the modifier.  Using a combination of 'Join'j and 'Edges->Rotate Edge'Ctrl+e and sliding vertices with gg, the area can be fixed but it's a bit of a Rubik's Cube to solve.  When the Solidy Modifier causes one of the editing operations to go bonkers, use Undo and try adding or rotating a different edge. 

I've posted a video to Vimeo that shows this in action https://vimeo.com/119991069